Problem: You have found the following ages (in years) of 6 sloths. Those sloths were randomly selected from the 36 sloths at your local zoo: $ 2,\enspace 3,\enspace 1,\enspace 1,\enspace 5,\enspace 7$ Based on your sample, what is the average age of the sloths? What is the standard deviation? You may round your answers to the nearest tenth.
Answer: Because we only have data for a small sample of the 36 sloths, we are only able to estimate the population mean and standard deviation by finding the sample mean $({\overline{x}})$ and sample standard deviation $({s})$ To find the sample mean , add up the values of all $6$ samples and divide by $6$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{6}} x_i}{{6}} $ $ {\overline{x}} = \dfrac{2 + 3 + 1 + 1 + 5 + 7}{{6}} = {3.2\text{ years old}} $ Find the squared deviations from the mean for each sample. Since we don't know the population mean, estimate the mean by using the sample mean we just calculated {1.44} + {0.04} + {4.84} + {4.84} + {3.24} + {14.44}} {{6 - 1}} $ {s^2} = \dfrac{{28.84}}{{5}} = {5.77\text{ years}^2} $ As you might guess from the notation, the sample standard deviation $({s})$ is found by taking the square root of the sample variance $({s^2})$ ${s} = \sqrt{{s^2}}$ $ {s} = \sqrt{{5.77\text{ years}^2}} = {2.4\text{ years}} $ We can estimate that the average sloth at the zoo is 3.2 years old. There is also a standard deviation of 2.4 years.